Theorem dfrals2 50625

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
dfrals2	⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

Proof of Theorem dfrals2

Step	Hyp	Ref	Expression
1		df-ral 3082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		impexp 456	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3	2	albii 1852	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	1, 3	bitr4i 281	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
5		df-rex 3092	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	4, 5	anbi12i 640	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		df-rals 50624	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
8		df-als 50623	. 2 ⊢ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  ∀∃wals 50621  ∀∃wrals 50622

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3082  df-rex 3092  df-als 50623  df-rals 50624

This theorem is used by:  rals-no-surprise  50642